# Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold

Fix $$n\geq 2$$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $$x\in \mathbb{H}^{n}$$ can be represented in polar coordinates $$x=(r, \theta)$$, and equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}(r)d\theta^{2},$$ where $$dr^{2}$$ is the standard metric on $$\mathbb{R}_{+}$$ and $$d\theta^{2}$$ is the standard metric on the sphere $$\mathbb{S}^{n-1}$$. Denote with $$\mu$$ the Riemannian measure on $$\mathbb{H}^{n}$$ and with $$\sigma$$ the Riemannian measure of co-dimension $$1$$ on hypersurfaces on $$\mathbb{H}^{n}$$. It is a known fact that $$\mathbb{H}^{n}$$ admits the Isoperimetric inequality $$\sigma(\partial \Omega)\geq f(\mu(\Omega)),$$ for all precompact open sets $$\Omega\subset \mathbb{H}^{n}$$ with smooth boundary, where $$f$$ is defined by $$f(v)=c_{1}\left\{\begin{array}{cl}v, &v\geq 1\\ v^{\frac{n-1}{n}}, &v\leq 1,\end{array}\right.$$ for some small constant $$c_{1}>0$$.

Let us fix some precompact open set $$U\subset \mathbb{H}^{n}$$ and consider $$\mathbb{H}^{n}\setminus U$$ as a manifold with boundary $$\partial U$$.

Now my question:

Does the same Isoperimetric inequality now hold for precompact open sets $$\Omega\subset \mathbb{H}^{n}\setminus U$$ with smooth boundary (and possibly a smaller constant $$c_{2}>0$$) and if yes, where can I find a reference for this statement?

Specified question:

Let $$U$$ be a "nice" precompact open set, for example $$U=B_{1}(o)$$ is an open ball of radius $$1$$ for some point $$o\in \mathbb{H}^{n}$$. Does the same Isoperimetric inequality now hold for precompact open sets $$\Omega\subset \mathbb{H}^{n}\setminus B_{1}(o)$$ with smooth boundary (and possibly a smaller constant $$c_{2}>0$$)?

Note that we assume that, when considering a precompact open set $$\Omega\subset \mathbb{H}^{n}\setminus U$$ with smooth boundary, we have $$\partial U\cap \partial \Omega=\emptyset$$.

If this is not known for the hyperbolic space $$\mathbb{H}^{n}$$, is a similar statement known for other Riemannian manifolds, for example $$\mathbb{R}^{n}$$?

• What do you mean by zero Neumann boundary condition in this context? Jul 5 at 13:59
• Assuming the answer to my first comment is that only the part of $\partial U$ on $\Omega$ is not counted, you cannot have a profile $f$ independent of $\Omega$ unless you impose some condition (bottlenecks could allow small perimeter and large volumes). Such an inequality does hold when $\Omega$ is convex. Jul 5 at 14:02
• Yes, it is meant as you assume it in your answer. Jul 5 at 14:33
• Do you have a reference for that statement, when $\Omega$ is convex? Jul 5 at 14:37
• I specified my question. Jul 5 at 14:52

Some results are known under some restrictions. First, no uniform inequality can hold if $$U$$ is unrestrained: it could take the shape of (a neighborhood of) a bottle, and the bottleneck will make it possible to have $$\Omega$$ with large volume and small boundary. A natural restriction is to consider convex $$U$$ (so that the double of $$\mathbb{H}^n$$ still is negatively curved in the metric sense).

To sum up: assuming $$U$$ is convex, we even have sharp bounds. When $$M=\mathbb{H}^n$$ the dimensions $$n=2,3,4$$ are covered; when $$M=\mathbb{R}^n$$, you have a weaker inequality (the isoperimetric function has the form $$f(v) = cv^{\frac{n-1}n}$$, there is no linear asymptotic) but known in all dimensions.

Added in edit: I think that an equality $$\sigma(\partial \Omega)\ge c\mu(\Omega)$$ is true in all dimensions whenever $$U$$ is convex, and that the method in our paper with Kuperberg mentionned below can be used to prove this. The more precise bound for small $$v$$, I am not sure.

Note that $$\mathbb{H}^n$$ can even be replaced by a simply connected manifold $$M$$ of sectional curvature bounded above by some $$\kappa\le 0$$. In this setting:

1. Choe proved in 2003 gave the sharp inequality when $$M=\mathbb{R}^n$$ and $$U$$ is a ball, as well as several other restricted cases: Relative isoperimetric inequality for domains outside a convex set. Archives Inequalities Appl 1 (2003): 241-250.

2. In 2007, the general case of a convex set $$U\subset \mathbb{R}^n$$ was obtained by Choe, Ghomi and Ritoré. The relative isoperimetric inequality outside convex domains in $$\mathbf{R}^n$$. Calculus of Variations and Partial Differential Equations 29.4 (2007): 421-429.

3. In 2006, Choe treated the case when $$M$$ has variable (nonpositive) curvature and dimension $$4$$. The double cover relative to a convex domain and the relative isoperimetric inequality. Journal of the Australian Mathematical Society 80.3 (2006): 375-382.

4. In dimension $$3$$, the same was achieved by Choe and Ritoré in 2007. The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds. J. für die reine und angewandte Mathematic (2007): 179-191. They also obtain the sharp inequality when $$\kappa=-1$$, in particular for $$M=\mathbb{H}^3$$.

5. With Kuperberg, we gave a new proof of Choe's result from 2006, treating dimensions $$2$$ and $$4$$ and including (sharp) results when $$M$$ has curvature bounded above by $$\kappa$$ (either $$0$$ or negative). The Cartan–Hadamard conjecture and the Little Prince. Revista Matemática Iberoamericana 35.4 (2019): 1195-1258. Our results can be used to tackle some finite union of convex sets (you need to ensure that geodesic rays reflecting on the boundary of $$U$$ can only bounce a bounded number of time).

I may have missed other relevant references, but you should catch them by looking at papers citing the above ones.

• Thank you very much for your answer! Going through these papers I was wondering if in the case of $\mathbb{H}^{n}$, the profile $f$ is exactly as it I have written it in my Post with $f$ being linear for large volumes? Jul 7 at 10:28
• @Shaq155: yes, more precisely the profile (over all convex $U$) is $f(v)=\frac12 f_0(2v)$ where $f_0$ is the profile of $\mathbb{H}^n$, which is linear for large $v$. This comes from the optimal case, a half ball against a flat wall. Jul 8 at 9:34
• @ Benoît Kloeckner: where exactly can I find the method which proves the linear asymptotic for large volumes in all dimensions in your paper with kuperberg or has there been made some progress by now? Dec 4 at 11:20
• @Shaq155: the relevant result in our paper with Kuperberg is Theorem 1.8; for this sharp bound, it is unfortunately necessary to digest most of the paper, which I admit is somewhat hard. But to prove a mere linear asymptotic, you can adapt the classical trick for the linear inequality in a Cartan-Hadamard manifold with curvature $\le -1$. The trick is to use Stokes equality on the gradient of either a Buseman function of the distance to a point. The adaptation for convex $U$ is to consider the double of $M\setminus U$ along $\partial U$, which is still CAT(-1). Dec 4 at 12:28